Question

Use u 41./+1/+1+0 1-1.j+U13-1-41,70 to approximate the solution of Laplace's equation at the interior points of the given region. Use symmetry when possible. (Assume u 0o= u(0,0). u(0, y) = 0, M2, y) = 100y, 0< y<3 u(x,0) = 0, UCX, 3) = 100x, 0

Answer 1

To approximate the solution of Laplace's equation at the interior points of the given region, we can use the method of relaxation. By discretizing the region into a grid and using the given boundary conditions, we can iteratively update the values of the interior points until the solution converges. Symmetry can be used to simplify the problem and reduce the number of calculations required.

Step-by-step explanation:

To approximate the solution of Laplace's equation at the interior points of the given region, we can use the method of relaxation. The given region has certain boundary conditions: u(0,0) = 0, u(0,y) = 0, u(2,y) = 100y, u(x,0) = 0, and u(x,3) = 100x. We can start by discretizing the region into a grid and assigning initial values to the interior points. Let's assume the grid has a spacing of h in both the x and y directions.

Using the given boundary conditions, we can update the values of the interior points iteratively until the solution converges. At each iteration, we can update the value of a point by taking the average of its neighboring points. We can continue this process until the solution reaches a desired level of accuracy.

By using symmetry, we can reduce the number of calculations required. For example, if we know the value of u at a point (x,y), we can determine the values at points (-x,y), (x,-y), and (-x,-y) using the given boundary conditions.